Roswitha Hofer

Distribution Properties of Generalized van der Corput-Halton Sequences and their Subsequences

We study the distribution properties of sequences which are a generalization of the well-known van der Corput–Halton sequences on one hand, and digital (T,s)-sequences on the other. In this paper, we give precise results concerning the distribution properties of such sequences in the s-dimensional unit cube. Moreover, we consider subsequences of the above-mentioned sequences and study their distribution properties. Additionally, we give discrepancy estimates for some special cases, including subsequences of van der Corput and van der Corput–Halton sequences.

ON HYBRID SEQUENCES BUILT FROM NIEDERREITER–HALTON SEQUENCES AND KRONECKER SEQUENCES

We discuss the distribution properties of hybrid sequences whose components stem from Niederreiter–Halton sequences on the one hand, and Kronecker sequences on the other. In this paper, we give necessary and sufficient conditions on the uniform distribution of such sequences, and derive a result regarding their discrepancy. We conclude with a short summary and a discussion of topics for future research.

A construction of digital (0,s)-sequences involving finite-row generator matrices

Abstract This paper presents a generalization of a construction method for digital ( 0 , s ) -sequences over F q introduced by Niederreiter which is based on hyperderivatives of polynomials over F q . Within this generalized concept, we are able to introduce direct constructions of finite-row digital ( 0 , s ) -sequences over arbitrary finite fields F q for arbitrary dimensions s ⩽ q . Previously, explicit examples of finite-row digital ( 0 , s ) -sequences have been known only for finite prime fields and for specific chosen dimensions. Further, this method furnishes additional insights into the structure of finite-row digital ( 0 , s ) -sequences and their generator matrices, and this approach permits shorter proofs for earlier interesting results on these sequences.

Halton-type sequences in rational bases in the ring of rational integers and in the ring of polynomials over a finite field

The aim of this paper is to generalize the well-known Halton sequences from integer bases to rational number bases and to translate this concept of Halton-type sequences in rational bases from the ring of integers to the ring of polynomials over a finite field. These two new classes of Halton-type sequences are low-discrepancy sequences. More exactly, the first class, based on the ring of integers, satisfies the discrepancy bounds that were recently obtained by Atanassov for the ordinary Halton sequence, and the second class, based on the ring of polynomials over a finite field, satisfies the discrepancy bounds that were recently introduced by Tezuka and by Faure & Lemieux for the generalized Niederreiter sequences.

On the Discrepancy of Halton–Kronecker Sequences

We study the discrepancy D N of sequences \(\left (\mathbf{z}_{n}\right )_{n\geq 1} = \left (\left (\mathbf{x}_{n},y_{n}\right )\right )_{n\geq 0} \in \left [\left.0,1\right.\right )^{s+1}\) where \(\left (\mathbf{x}_{n}\right )_{n\geq 0}\) is the s-dimensional Halton sequence and \(\left (y_{n}\right )_{n\geq 1}\) is the one-dimensional Kronecker-sequence \(\left (\left \{n\alpha \right \}\right )_{n\geq 1}\). We show that for α algebraic we have \(ND_{N} = \mathcal{O}\left (N^{\varepsilon }\right )\) for all ɛ > 0. On the other hand, we show that for α with bounded continued fraction coefficients we have \(ND_{N} = \mathcal{O}\left (N^{\frac{1} {2} }(\log N)^{s}\right )\) which is (almost) optimal since there exist α with bounded continued fraction coefficients such that \(ND_{N} = \Omega \left (N^{\frac{1} {2} }\right )\).

On the discrepancy of two-dimensional perturbed Halton--Kronecker sequences and lacunary trigonometric products

We consider distribution properties of two-dimensional hybrid sequences

Generalized Hofer-Niederreiter sequences and their discrepancy from an (u, e, s)-point of view

(z_k)_{k}

A Finite-Row Scrambling of Niederreiter Sequences

in the unit square of the form

A metric result for special sequences related to the Halton sequences

z_k=(\{k\alpha\},x_k)

Kronecker–Halton sequences in Fp((X−1))

, where